Mental math trick for squared 20s and squared teen numbers? I'm working on a pullup progression workout. I'd like to mental math how many pull-ups done in one session. I do decrements from $20s$; e.g., $20,19,18,17,16,15,14,13,12,11,10$. This is like the summation version of a factorial, the nth triangular number:

$\frac{(n^2+n)}{2}$
n = 20
(400+20)/2 = 210
n = 9
(81+9)/2 = 45
$\sum_{n=10}^{20} n = 210-45 = 165$
So $165$ pull-ups in one workout.
But each week I usually increase the amount, and end earlier; e.g., $21,20,19,18,17,16,15,14,13,12$. 

I'm not sure if this is appropriate for this SE, but I'd like to figure out the best way using mental math. The goal is to do 50 continuous pullups, then maybe 80...
Anyway, I want to learn a method to the mathness for 9 or 10 sets. (Sometimes I don't feel like doing the 10th set.) I think once I'm able to do 29 decrement for 9 sets, I'll be able to do 50 continuous pullups. Here's a list of n²:
21² = 441
22² = 484
23² = 529
24² = 576
25² = 625
26² = 676
27² = 729
28² = 784
29² = 841
Mental math for squared 20s?
11² = 121
12² = 144
13² = 169
14² = 196
15² = 225
16² = 256
17² = 289
18² = 324
19² = 361
Mental math for squared teens?
Note how these 18 numbers could be memorized. Although, the 18 results (total pull-ups) would be better memorize in the first place:

I also just want to do the mental math rather than memorize. So if you know a better way than the two-step triangular number mental math (which is more like three-steps if a subtraction method is worked in), that'd be better to know.

 A: You can very easily calculate the squares up to $75^2$ if you've memorised the squares up to $25^2=625$.  I'll give an example.
To square $37$, first square the difference from $37$ to $50$, that is $13^2=169$.  Then add as many hundreds as $37$ differs from $25$, that is, add $1200$.  Add both to obtain $37^2=169+1200=1369$.  
Another one: For $56^2$ first square $50-56$ to obtain $36$ and add $56-25=31$ hundreds to obtain $56^2=16+3100=3136$.
Proof: Expand
$$a^2=(50-a)^2+100(a-25).$$
For numbers ending in $5$ it's even easier.  Example: to square $75$, calculate $7\cdot8$ and concatenate $25$ to obtain $75^2=5625$.  For $125^2$ do $12\cdot13=156$ and concatenate $25$ to get $125^2=15625$.
Proof: Left to the reader.
A: Teen numbers
11*11 = 121 (memorize)
12*12 = 144 (memorize)
13*13 = (1*1).(3+3).(3*3) = 1.6.9 = 169
14*14 = (1*1).(4+4).(4*4) = 1.8.(1)6 = 196 (carry one from 16 to 8)
15*15 = (1*1).(5+5).(5*5) = 1.(1)0.(2)5 = 225 (carry 1 higher, & carry 2 higher)
16*16 = (1*1).(6+6).(6*6) = 1.(1)2.(3)6 = 256 (carry 1 higher, & carry 3 higher: 3+2)
etc.
20s method
XY (two-digit number), want (XY)²
X² = J
Y² = K
Concatenate J and K for JK.
(X*Y*2, and put zero on end) = L
(XY)² = JK+L
Example
26² 
→ 4=J
→ 36=K
→ JK = 436 
→ 2*6*2 = 24 → 240
436+240
= 676 = 26²
Method 2
https://medium.com/i-math/find-perfect-squares-mentally-with-this-trick-b8fa2c116d73 

Supplementary


*

*Faster mental calculations

*https://www.soa.org/news-and-publications/newsletters/compact/2013/april/com-2013-iss47/mental-mathbe-there-or-b2-squaring-two-digit-numbers/

*https://en.wikibooks.org/wiki/Mental_Math
